A report on research into teaching the topic Complex Numbers, within the NSW syllabus
A review of the syllabus documents (NESA 2017), linked above, reveals a topic which focuses on the abstract. No specific applications outside pure mathematics are mentioned. This presents some challenges, but also the opportunity to consider numbers and mathematical structures in a more formal way, while reviewing some of the foundations. Introducing the new number i and the remarkable way such a simple idea can open huge new vistas in mathematics can be a path to expanding students' ideas about what mathematics is and what mathematical thinking involves at its best and most exciting.
What the syllabus outlines is the algebraic, geometric and trigonometric aspects of complex numbers. Their power is they unify these areas which have been somewhat distinct till this stage in the student's journey through mathematics. There have been opportunities where they intersect. Pythagoras Theorem keeps cropping up through the years in new contexts. Quadratics and cubics relate to area and space. These two topics are perhaps the most ancient serious mathematics, developed well over 1000 years before Pythagoras and classical Greek culture as formal, written ideas. We have collected some ideas and resources helping to illuminate this background, this journey through mathematics, and perhaps to reflect on not just our own journeys but the collective journey that has led us to this point over the millenia, see the 'Numbers' link below.
Perhaps we can see the invention of counting (millenia earlier than that) as the first mathematics (Everett 2017). Or perhaps the geometry which is clearly possible without numbers at all. Here we offer some reading, including a fascinating examination of the last two statements, as a way to help leverage this topic for the purposes outlined above. See the 'Books' link below for material suitable for teachers as background, or for offering to students at various levels of mathematical understanding. This material is very relevant to teachers and students seeking to place Complex Numbers as the next enhancement of that number system, which we have invented to help us build models and describe the world. Not the last one, but it is the last we will meet formally in high school. At this stage they are the keystone that links the strands, but that is a very formal thing, and it was not the original motivation. It was something discovered later.
The principle that meaningful lessons can foster positive attitudes, improve lesson enjoyment and enhance student learning (Whittle, 2007) is, quite properly, at the heart of much lesson planning. The challenge here is how to place Complex Numbers in a way that is meaningful in applied mathematics, apart from the pure mathematical ways outlined above. They have remained part of mathematics (despite the doubts of the first mathematicians who used them) for almost 500 years now. This suggests they must have some practical applications, contemporary usage on the cutting edge of physics aside (that application is amazing, but tough (Penrose, 2004)). We looked at where else they are used, see the 'Applications' link below for a variety of resources and commentary on the outcome. Many of the places that complex numbers help solve certain algebraic situations require too much setting up and too much mathematical knowledge from beyond this syllabus. Johnson (2014) suggests that complex numbers are normally used when some quantity has a magnitude as well as a phase, and gives practical examples in physics and engineering which include electrical circuits, vibrations and resonances, optical wave interference and quantum mechanical wave functions.
We propose introducing the topic with solutions to cubic equations, Bagni (2010) has looked closely at the advantages of this approach. See the 'Intro-lesson' link below for details. Can we find an approach that offers some practical purpose, some motivation for "just making it up" other than stepping back and saying "isn't it beautiful". The introduction of i is hard. We (teachers) have hammered the impossibility of taking the square root of a negative number over and over again. It was the rule. This may have been a mistake, but we have. But in this respect our students are just like the mathematicians of 1530s.
Bagni (2000) considered this, and suggests that the historical order could inform our teaching. He tested this on students. The quadratic setting, with the final answer as i just hits rejection, this is not allowed, i is not the solution, there is no solution. But going via a complex number to arrive at a real solution, a solution that can be seen to be correct independently, is much more believable. The historical narrative shows the resistance, Bagni confirms the effect on students. He proposes using the historical artefacts of Bombelli's publication that first clearly laid out the system.
The history of our development of difficult ideas can help understand the motivations behind them, and the most practical ideas driving them. As mathematicians we have often refined and justified the formal structures with the benefit of hindsight, and in some ways this makes the concepts clearer. But it can obscure the relationships to the wider world that were the primary reason for the invention, or at least for persisting with it beyond the joy of recognising something elegant and compelling in its own right. How to approach teaching calculus can be considered in this respect. Should we first look at limits and the 19th century formal derivations, or should we start with the original motivation, considering calculus first as the language for describing change over time, and especially for understanding the motion of objects under the influence of gravity? To understand better the place of complex numbers we researched their historical development, and the outcome and resources are shared under the 'History' link below.
Penrose (2004), Lockhart (2017, 2002), Schrödinger (1944), Gleick (1987) and many more all consider, in their very different ways, this relationship between the beauty of the pure idea and the way it builds the language we use to describe and understand our world. Time (Butler & Messel 1965) has been in my bookshelf for 40 years, a series of lectures and activities for a summer camp at Sydney University for "year 4" (now 10 or 11) high school students. The purpose of the camp was to inspire, with a connection between the beauty of the mathematical ideas and the surprising but very real outcomes they describe. In their case the challenge was time and relativity. They included quite a range of activities and ideas not directly related to the topic.